1. I unhesitatingly answer the question in the title: not only can one play with a tense hand, but it is even a must!
1.1. But only if you tense it in an intelligent and efficient manner - first of all, not permanently, but selectively (i.e. contracting only certain muscles) and by impulses. In other words, while playing, the hand should be both a little contracted and a little relaxed.
1.2. The hand and arm totally relaxed, inert can only be called a few kilos of muscles around a few bones and as such cannot accomplish any task, even the simplest. Consequently, it is impossible to play the piano in this manner. The rule, taught to students for over two hundred years, according to which one must play with a relaxed hand, is not precise enough. It may lead to confusion, and that is what unfortunately happens very frequently!
“[...] a superior member weights an average of 3.6 kilos (with the shoulder).
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Czesław Sielużycki - “The pianist’s hand” p. 120, PWM, Cracow, 1982